3.706 \(\int \frac{x^{14}}{\left (a+b x^6\right )^2 \sqrt{c+d x^6}} \, dx\)

Optimal. Leaf size=141 \[ -\frac{\sqrt{a} (3 b c-2 a d) \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{6 b^2 (b c-a d)^{3/2}}+\frac{a x^3 \sqrt{c+d x^6}}{6 b \left (a+b x^6\right ) (b c-a d)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{d} x^3}{\sqrt{c+d x^6}}\right )}{3 b^2 \sqrt{d}} \]

[Out]

(a*x^3*Sqrt[c + d*x^6])/(6*b*(b*c - a*d)*(a + b*x^6)) - (Sqrt[a]*(3*b*c - 2*a*d)
*ArcTan[(Sqrt[b*c - a*d]*x^3)/(Sqrt[a]*Sqrt[c + d*x^6])])/(6*b^2*(b*c - a*d)^(3/
2)) + ArcTanh[(Sqrt[d]*x^3)/Sqrt[c + d*x^6]]/(3*b^2*Sqrt[d])

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Rubi [A]  time = 0.441343, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{\sqrt{a} (3 b c-2 a d) \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{6 b^2 (b c-a d)^{3/2}}+\frac{a x^3 \sqrt{c+d x^6}}{6 b \left (a+b x^6\right ) (b c-a d)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{d} x^3}{\sqrt{c+d x^6}}\right )}{3 b^2 \sqrt{d}} \]

Antiderivative was successfully verified.

[In]  Int[x^14/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

(a*x^3*Sqrt[c + d*x^6])/(6*b*(b*c - a*d)*(a + b*x^6)) - (Sqrt[a]*(3*b*c - 2*a*d)
*ArcTan[(Sqrt[b*c - a*d]*x^3)/(Sqrt[a]*Sqrt[c + d*x^6])])/(6*b^2*(b*c - a*d)^(3/
2)) + ArcTanh[(Sqrt[d]*x^3)/Sqrt[c + d*x^6]]/(3*b^2*Sqrt[d])

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Rubi in Sympy [A]  time = 49.5873, size = 122, normalized size = 0.87 \[ - \frac{\sqrt{a} \left (2 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{x^{3} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{6}}} \right )}}{6 b^{2} \left (a d - b c\right )^{\frac{3}{2}}} - \frac{a x^{3} \sqrt{c + d x^{6}}}{6 b \left (a + b x^{6}\right ) \left (a d - b c\right )} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{d} x^{3}}{\sqrt{c + d x^{6}}} \right )}}{3 b^{2} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**14/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)

[Out]

-sqrt(a)*(2*a*d - 3*b*c)*atanh(x**3*sqrt(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**6)))/
(6*b**2*(a*d - b*c)**(3/2)) - a*x**3*sqrt(c + d*x**6)/(6*b*(a + b*x**6)*(a*d - b
*c)) + atanh(sqrt(d)*x**3/sqrt(c + d*x**6))/(3*b**2*sqrt(d))

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Mathematica [A]  time = 0.360359, size = 135, normalized size = 0.96 \[ \frac{\frac{a b x^3 \sqrt{c+d x^6}}{\left (a+b x^6\right ) (b c-a d)}+\frac{\sqrt{a} (2 a d-3 b c) \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{(b c-a d)^{3/2}}+\frac{2 \log \left (\sqrt{d} \sqrt{c+d x^6}+d x^3\right )}{\sqrt{d}}}{6 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^14/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

((a*b*x^3*Sqrt[c + d*x^6])/((b*c - a*d)*(a + b*x^6)) + (Sqrt[a]*(-3*b*c + 2*a*d)
*ArcTan[(Sqrt[b*c - a*d]*x^3)/(Sqrt[a]*Sqrt[c + d*x^6])])/(b*c - a*d)^(3/2) + (2
*Log[d*x^3 + Sqrt[d]*Sqrt[c + d*x^6]])/Sqrt[d])/(6*b^2)

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Maple [F]  time = 0.075, size = 0, normalized size = 0. \[ \int{\frac{{x}^{14}}{ \left ( b{x}^{6}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^14/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)

[Out]

int(x^14/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{14}}{{\left (b x^{6} + a\right )}^{2} \sqrt{d x^{6} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/((b*x^6 + a)^2*sqrt(d*x^6 + c)),x, algorithm="maxima")

[Out]

integrate(x^14/((b*x^6 + a)^2*sqrt(d*x^6 + c)), x)

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Fricas [A]  time = 0.595846, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/((b*x^6 + a)^2*sqrt(d*x^6 + c)),x, algorithm="fricas")

[Out]

[1/24*(4*sqrt(d*x^6 + c)*a*b*sqrt(d)*x^3 + ((3*b^2*c - 2*a*b*d)*x^6 + 3*a*b*c -
2*a^2*d)*sqrt(d)*sqrt(-a/(b*c - a*d))*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^1
2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^6 + a^2*c^2 - 4*((b^2*c^2 - 3*a*b*c*d + 2*a^2*d^
2)*x^9 - (a*b*c^2 - a^2*c*d)*x^3)*sqrt(d*x^6 + c)*sqrt(-a/(b*c - a*d)))/(b^2*x^1
2 + 2*a*b*x^6 + a^2)) + 4*((b^2*c - a*b*d)*x^6 + a*b*c - a^2*d)*log(-2*sqrt(d*x^
6 + c)*d*x^3 - (2*d*x^6 + c)*sqrt(d)))/(((b^4*c - a*b^3*d)*x^6 + a*b^3*c - a^2*b
^2*d)*sqrt(d)), 1/24*(4*sqrt(d*x^6 + c)*a*b*sqrt(-d)*x^3 + ((3*b^2*c - 2*a*b*d)*
x^6 + 3*a*b*c - 2*a^2*d)*sqrt(-d)*sqrt(-a/(b*c - a*d))*log(((b^2*c^2 - 8*a*b*c*d
+ 8*a^2*d^2)*x^12 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^6 + a^2*c^2 - 4*((b^2*c^2 - 3*a
*b*c*d + 2*a^2*d^2)*x^9 - (a*b*c^2 - a^2*c*d)*x^3)*sqrt(d*x^6 + c)*sqrt(-a/(b*c
- a*d)))/(b^2*x^12 + 2*a*b*x^6 + a^2)) + 8*((b^2*c - a*b*d)*x^6 + a*b*c - a^2*d)
*arctan(sqrt(-d)*x^3/sqrt(d*x^6 + c)))/(((b^4*c - a*b^3*d)*x^6 + a*b^3*c - a^2*b
^2*d)*sqrt(-d)), 1/12*(2*sqrt(d*x^6 + c)*a*b*sqrt(d)*x^3 - ((3*b^2*c - 2*a*b*d)*
x^6 + 3*a*b*c - 2*a^2*d)*sqrt(d)*sqrt(a/(b*c - a*d))*arctan(1/2*((b*c - 2*a*d)*x
^6 - a*c)/(sqrt(d*x^6 + c)*(b*c - a*d)*x^3*sqrt(a/(b*c - a*d)))) + 2*((b^2*c - a
*b*d)*x^6 + a*b*c - a^2*d)*log(-2*sqrt(d*x^6 + c)*d*x^3 - (2*d*x^6 + c)*sqrt(d))
)/(((b^4*c - a*b^3*d)*x^6 + a*b^3*c - a^2*b^2*d)*sqrt(d)), 1/12*(2*sqrt(d*x^6 +
c)*a*b*sqrt(-d)*x^3 - ((3*b^2*c - 2*a*b*d)*x^6 + 3*a*b*c - 2*a^2*d)*sqrt(-d)*sqr
t(a/(b*c - a*d))*arctan(1/2*((b*c - 2*a*d)*x^6 - a*c)/(sqrt(d*x^6 + c)*(b*c - a*
d)*x^3*sqrt(a/(b*c - a*d)))) + 4*((b^2*c - a*b*d)*x^6 + a*b*c - a^2*d)*arctan(sq
rt(-d)*x^3/sqrt(d*x^6 + c)))/(((b^4*c - a*b^3*d)*x^6 + a*b^3*c - a^2*b^2*d)*sqrt
(-d))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**14/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.253873, size = 482, normalized size = 3.42 \[ \frac{1}{6} \, c^{2}{\left (\frac{{\left (3 \, a b c - 2 \, a^{2} d\right )} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{6}}}}{\sqrt{a b c - a^{2} d}}\right )}{{\left (b^{3} c^{3}{\rm sign}\left (x\right ) - a b^{2} c^{2} d{\rm sign}\left (x\right )\right )} \sqrt{a b c - a^{2} d}} + \frac{a \sqrt{d + \frac{c}{x^{6}}}}{{\left (b^{2} c^{2}{\rm sign}\left (x\right ) - a b c d{\rm sign}\left (x\right )\right )}{\left (b c + a{\left (d + \frac{c}{x^{6}}\right )} - a d\right )}} - \frac{2 \, \arctan \left (\frac{\sqrt{d + \frac{c}{x^{6}}}}{\sqrt{-d}}\right )}{b^{2} c^{2} \sqrt{-d}{\rm sign}\left (x\right )}\right )} - \frac{{\left (3 \, a b c \sqrt{-d} \arctan \left (\frac{a \sqrt{d}}{\sqrt{a b c - a^{2} d}}\right ) - 2 \, a^{2} \sqrt{-d} d \arctan \left (\frac{a \sqrt{d}}{\sqrt{a b c - a^{2} d}}\right ) - 2 \, \sqrt{a b c - a^{2} d} b c \arctan \left (\frac{\sqrt{d}}{\sqrt{-d}}\right ) + 2 \, \sqrt{a b c - a^{2} d} a d \arctan \left (\frac{\sqrt{d}}{\sqrt{-d}}\right ) + \sqrt{a b c - a^{2} d} a \sqrt{-d} \sqrt{d}\right )}{\rm sign}\left (x\right )}{6 \,{\left (\sqrt{a b c - a^{2} d} b^{3} c \sqrt{-d} - \sqrt{a b c - a^{2} d} a b^{2} \sqrt{-d} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/((b*x^6 + a)^2*sqrt(d*x^6 + c)),x, algorithm="giac")

[Out]

1/6*c^2*((3*a*b*c - 2*a^2*d)*arctan(a*sqrt(d + c/x^6)/sqrt(a*b*c - a^2*d))/((b^3
*c^3*sign(x) - a*b^2*c^2*d*sign(x))*sqrt(a*b*c - a^2*d)) + a*sqrt(d + c/x^6)/((b
^2*c^2*sign(x) - a*b*c*d*sign(x))*(b*c + a*(d + c/x^6) - a*d)) - 2*arctan(sqrt(d
+ c/x^6)/sqrt(-d))/(b^2*c^2*sqrt(-d)*sign(x))) - 1/6*(3*a*b*c*sqrt(-d)*arctan(a
*sqrt(d)/sqrt(a*b*c - a^2*d)) - 2*a^2*sqrt(-d)*d*arctan(a*sqrt(d)/sqrt(a*b*c - a
^2*d)) - 2*sqrt(a*b*c - a^2*d)*b*c*arctan(sqrt(d)/sqrt(-d)) + 2*sqrt(a*b*c - a^2
*d)*a*d*arctan(sqrt(d)/sqrt(-d)) + sqrt(a*b*c - a^2*d)*a*sqrt(-d)*sqrt(d))*sign(
x)/(sqrt(a*b*c - a^2*d)*b^3*c*sqrt(-d) - sqrt(a*b*c - a^2*d)*a*b^2*sqrt(-d)*d)